I totally agree with @Darwin, many of us buy as much house as we can afford and don’t think about living expenses (fixing things, remodeling, etc.), so if you are able to save a bit for that, it’s a nice cushion to have.
I think the general rule is that your house payment shouldn’t be more than 40% of your take home pay, I would encourage you to try to go even lower because that has always been a tight number for us…maybe as low as 25–30%, if possible. I’m not sure how realistic or possible that is for you.
Also, definitely put 20% down if you can, like @Snoopy said, but I would say if you can handle a 15 year loan, fixed rate, try to get that and then paying extra on top, if possible.
If you don’t like your real estate agent, get a new one, you are in no way bound to them just because you saw a few houses with them.
Definitely get a home inspection and a pest inspection (we have termite problems here), those are well worth the money spent.
And remember that everything is negotiable! Have fun and enjoy the process. :)
Oh, try to get prequalified for a loan right now while you are looking.